Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 5

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos x}{\frac{y}{\sinh y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (cos x) (/ y (sinh y))))

C

double code(double x, double y) {
	return cos(x) / (y / sinh(y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) / (y / sinh(y))
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) / (y / Math.sinh(y));
}

Python

def code(x, y):
	return math.cos(x) / (y / math.sinh(y))

Julia

function code(x, y)
	return Float64(cos(x) / Float64(y / sinh(y)))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) / (y / sinh(y));
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 55.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\cos x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
4. Step-by-step derivation

   1. associate-*r/ 55.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\cos x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right)}{y}} \]

   2. *-commutative 55.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \cos x\right)}}{y} \]

   3. associate-*r* 55.2%

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) \cdot \cos x}}{y} \]

   4. associate-/l* 55.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) \cdot \frac{\cos x}{y}} \]

   5. *-commutative 55.2%

      \[\leadsto \color{blue}{\left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 0.5\right)} \cdot \frac{\cos x}{y} \]

   6. metadata-eval 55.2%

      \[\leadsto \left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{\cos x}{y} \]

   7. associate-/l* 55.2%

      \[\leadsto \color{blue}{\frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 1}{2}} \cdot \frac{\cos x}{y} \]

   8. *-rgt-identity 55.2%

      \[\leadsto \frac{\color{blue}{e^{y} - \frac{1}{e^{y}}}}{2} \cdot \frac{\cos x}{y} \]

   9. rec-exp 55.2%

      \[\leadsto \frac{e^{y} - \color{blue}{e^{-y}}}{2} \cdot \frac{\cos x}{y} \]

   10. sinh-def 99.9%

       \[\leadsto \color{blue}{\sinh y} \cdot \frac{\cos x}{y} \]

   11. *-commutative 99.9%

       \[\leadsto \color{blue}{\frac{\cos x}{y} \cdot \sinh y} \]

   12. associate-/r/ 100.0%

       \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]
6. Add Preprocessing

Alternative 2: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.00000001:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.00000001) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.00000001) {
		tmp = cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 1.00000001d0) then
        tmp = cos(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 1.00000001) {
		tmp = Math.cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.00000001:
		tmp = math.cos(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.00000001)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.00000001)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 1.00000001], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.0000000099999999

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.8%

      \[\leadsto \color{blue}{\cos x} \]

   if 1.0000000099999999 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 74.3%

         \[\leadsto \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{y} \cdot 0.5} \]

      2. associate-*l/ 74.3%

         \[\leadsto \color{blue}{\frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 0.5}{y}} \]

      3. metadata-eval 74.3%

         \[\leadsto \frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}}}{y} \]

      4. associate-/l* 74.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 1}{2}}}{y} \]

      5. *-rgt-identity 74.3%

         \[\leadsto \frac{\frac{\color{blue}{e^{y} - \frac{1}{e^{y}}}}{2}}{y} \]

      6. rec-exp 74.3%

         \[\leadsto \frac{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}{y} \]

      7. sinh-def 74.4%

         \[\leadsto \frac{\color{blue}{\sinh y}}{y} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 4: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cos x))

C

double code(double x, double y) {
	return cos(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x);
}

Python

def code(x, y):
	return math.cos(x)

Julia

function code(x, y)
	return cos(x)
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x);
end

Wolfram

code[x_, y_] := N[Cos[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 49.0%

   \[\leadsto \color{blue}{\cos x} \]
4. Add Preprocessing

Alternative 5: 29.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 41.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
4. Step-by-step derivation

   1. *-commutative 41.3%

      \[\leadsto \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{y} \cdot 0.5} \]

   2. associate-*l/ 41.3%

      \[\leadsto \color{blue}{\frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 0.5}{y}} \]

   3. metadata-eval 41.3%

      \[\leadsto \frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}}}{y} \]

   4. associate-/l* 41.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot 1}{2}}}{y} \]

   5. *-rgt-identity 41.3%

      \[\leadsto \frac{\frac{\color{blue}{e^{y} - \frac{1}{e^{y}}}}{2}}{y} \]

   6. rec-exp 41.3%

      \[\leadsto \frac{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}{y} \]

   7. sinh-def 67.5%

      \[\leadsto \frac{\color{blue}{\sinh y}}{y} \]

5. Simplified 67.5%

   \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

6. Taylor expanded in y around 0 29.7%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))